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1. \frac{d}{dx}\left[- \sin{\left(2 t - x \right)}\right]

   Differentiate 2 times, one derivative at a time.

2. - \frac{\partial}{\partial x} \sin{\left(2 t - x \right)}

   Constant multiple rule: pull the constant out.

3. - \cos{\left(2 t - x \right)} \frac{\partial}{\partial x} \left(2 t - x\right)

   Chain rule: (sin u)′ = cos u, times u′ where u = 2 t - x.

4. - \left(\frac{d}{d x} 2 t + \frac{d}{d x} \left(- x\right)\right) \cos{\left(2 t - x \right)}

   Sum rule: differentiate term by term.

5. - \left(\frac{d}{d x} 2 t - \frac{d}{d x} x\right) \cos{\left(2 t - x \right)}

   Constant multiple rule: pull the constant out.

6. - \left(\frac{d}{d x} 2 t - 1\right) \cos{\left(2 t - x \right)}

   d/dx of x is 1.

7. \cos{\left(2 t - x \right)}

   The derivative of a constant is 0.

8. f^{(1)}(x) = \cos{\left(2 t - x \right)}

   That is derivative number 1.

9. - \sin{\left(2 t - x \right)} \frac{\partial}{\partial x} \left(2 t - x\right)

   Chain rule: (cos u)′ = −sin u, times u′ where u = 2 t - x.

10. - \left(\frac{d}{d x} 2 t + \frac{d}{d x} \left(- x\right)\right) \sin{\left(2 t - x \right)}

    Sum rule: differentiate term by term.

11. - \left(\frac{d}{d x} 2 t - \frac{d}{d x} x\right) \sin{\left(2 t - x \right)}

    Constant multiple rule: pull the constant out.

12. - \left(\frac{d}{d x} 2 t - 1\right) \sin{\left(2 t - x \right)}

    d/dx of x is 1.

13. \sin{\left(2 t - x \right)}

    The derivative of a constant is 0.

14. f^{(2)}(x) = \sin{\left(2 t - x \right)}

    That is derivative number 2.